December 7, 2008

This post is about simulated annealing (SA), before the age of quantum computers. Its purpose is merely to quote a cute formula that gives an upper bound to the number of iterations that classical SA needs, in order to find an absolute minimum of an energy function , allowing a certain probability of failure. Here is the formula:

Eq.(1) is THE BOUNDER TO BEAT by quantum computers, if they are to do SA faster than classical computers. (a bounder is something that bounds; it’s also a cad or ruffian in British slang).

References Som07, Som08, and Woc08 have already given us several quantum computer algorithms that beat this bounder. Their algorithms give rather than . This will be significant for tough NP-complete problems where is very small. One of my New Year’s resolutions is to come up with my own version of a quantum algorithm that beats Eq.(1).

In the remainder of this post, I will define the variables used in Eq.(1). A proof of Eq.(1) may be found in Appendix A of Ref. Som07.

Consider the Markov chain that underlies SA.

I will underline symbols that represent random variables. A Markov chain is a Bayesian network , where all the random variables , where , have the same set of possible values (a.k.a. states).

Let be the number of elements in . Let be the transition matrix of the Markov chain at time . Let have eigenvalues . Let be the “Markov chain spectral gap”.

Suppose we have an energy function . Let , . Let be the “energy gap”.

Let be the failure probability.

One can show that the “final time” (i.e. total number of links in the chain, total number of SA iterations) is upper bounded by Eq.(1).References: